Simple New Keynesian Model without Capital

Lawrence J. Christiano

March 10, 2020

Objective

Review the foundations of the basic New Keynesian model without capital.

I Work out several basic results (dichotomy when prices flexible).

Derive the Equilibrium Conditions.

I Small number of equations and a small number of variables, which summarize

everything about the model (optimization, market clearing, gov’t policy, etc.).

Study some of the properties of the model.

I Do this using Dynare and ‘pencil and paper’ methods.

Outline

The model:

I Individual agents: their objectives, what they take as given, what they choose.

F Households, final good firms, intermediate good firms.

I Economy-wide restrictions:

F Market clearing conditions.

F Relationship between aggregate output and aggregate factors of production,

aggregate price level and individual prices.

Properties of Equilibrium:

I Classical Dichotomy - when prices flexible monetary policy irrelevant for real

variables.

I Monetary policy essential to determination of all variables when prices sticky.

Households

Households’ problem.

Concept of Consumption Smoothing.

Households

There are many identical households.

The problem of the typical (’representative’) household:

maxE0

∞∑t=0

βt

(logCt − exp (τt)

N1+ϕt

1 + ϕ

),

s.t. PtCt + Bt+1

≤WtNt + Rt−1Bt

+Profits net of government transfers and taxest .

Here, Bt denote the beginning-of-period t stock of bonds held by the

household.

Law of motion of the shock to preferences:

τt = λτt−1 + ετt

the preference shock is in the model for pedagogic purposes only, it is not an

interest shock from an empirical point of view.

Household First Order Conditions

The household first order conditions: EulerEquation

1

Ct= βEt

1

Ct+1

Rt

πt+1(5)

eτtCtNϕt =

Wt

Pt.

All equations are derived by expressing the household problem in

Lagrangian form, substituting out the multiplier on budget constraint and

rearranging.

Consumption Smoothing

Later, we’ll see that consumption smoothing is an important principle for

understanding the role of monetary policy in the New Keynesian model.

Consumption smoothing is a characteristic of households’ consumption

decision when they expect a change in income and the interest rate is not

expected to change.

I Peoples’ current period consumption increases by the amount that can,

according to their budget constraint, be maintained indefinitely.

Consumption Smoothing: Example

Problem:

maxc1,c2 log (c1) + βlog (c2)

subject to : c1 + B1 ≤ y1 + rB0

c2 ≤ rB1 + y2.

where y1 and y2 are (given) income and, after imposing equality (optimality)

and substituting out for B1,

c1 +c2

r= y1 +

y2

r+ rB0,

1

c1= βr

1

c2,

second equation is fonc for B1.

Suppose βr = 1 (this happens in ’steady state’, see later):

c1 =y1 + y2

r

1 + 1r

+r

1 + 1r

B0

Consumption Smoothing: Example, cnt’d

Solution to the problem:

c1 =y1 + y2

r

1 + 1r

+r

1 + 1r

B0.

Consider three polar cases:

I temporary change in income: ∆y1 > 0 and ∆y2 = 0 =⇒ ∆c1 = ∆c2 = ∆y1

1+ 1r

I permanent change in income: ∆y1 = ∆y2 > 0 =⇒ ∆c1 = ∆c2 = ∆y1

I future change in income: ∆y1 = 0 and ∆y2 > 0 =⇒ ∆c1 = ∆c2 =∆y2r

1+ 1r

Common feature of each example:

I When income rises, then - assuming r does not change - c1 increases by an

amount that can be maintained into the second period: consumption

smoothing.

Final Goods Production

A homogeneous final good is produced using the following (Dixit-Stiglitz)

production function:

Yt =

[∫ 1

0

Yε−1ε

i,t di

] εε−1

.

Each intermediate good,Yi,t , is produced by a monopolist using the following

production function:

Yi,t = eatNi,t , at ∼ exogenous shock to technology.

Final Good Producers

Competitive firms:

I maximize profits

PtYt −∫ 1

0

Pi,tYi,tdj ,

subject to Pt , Pi,t given, all i ∈ [0, 1] , and the technology:

Yt =

[∫ 1

0

Yε−1ε

i,t dj

] εε−1

.

Foncs:

Yi,t = Yt

(Pt

Pi,t

)ε→

”aggregate price index”︷ ︸︸ ︷Pt =

(∫ 1

0

P(1−ε)i,t di

) 11−ε

Intermediate Good Producers

The i th intermediate good is produced by a monopolist.

Demand curve for i th monopolist:

Yi,t = Yt

(Pt

Pi,t

)ε.

Production function:

Yi,t = eatNi,t , at ˜ exogenous shock to technology.

Calvo Price-Setting Friction: rotemberg

Pi,t =

Pt with probability 1− θPi,t−1 with probability θ

.

Marginal Cost of Production

An important input into the monopolist’s problem is its marginal cost: derive

MCt =dCost

dOutput=

dCostdWorkerdOutputdWorker

=(1− ν)Wt

eat

=(1− ν) eτtCtN

ϕt

eatPt

after substituting out for the real wage from the household intratemporal

Euler equation.

The tax rate, ν, represents a subsidy to hiring labor, financed by a lump-sum

government tax on households.

Firm’s job is to set prices whenever it has the opportunity to do so.

I It must always satisfy whatever demand materializes at its posted price.

Present Discounted Value of Intermediate Good Revenues

i th intermediate good firm’s objective:

E it

∞∑j=0

βj υt+j

period t+j profits sent to household︷ ︸︸ ︷ revenues︷ ︸︸ ︷Pi,t+jYi,t+j −

total cost︷ ︸︸ ︷Pt+jst+jYi,t+j

υt+j - Lagrange multiplier on household budget constraint

ExplainDiscounting

Here, E it denotes the firm’s expectation over future variables, where the i

notation indicates that the expectation also takes into account the firm’s

idiosyncratic Calvo price-setting uncertainty.

Also, st denotes MCt/Pt .

Firm that Can Change Its Price (‘Marginal Price Setter’)

The 1− θ firms that are can set their price at t do so as follows:

Pt = Et

∞∑j=0

ωt,jε

ε− 1MCt+j , Et

∞∑j=0

ωt,j = 1

where absence of i index means expectation is only over aggregate

uncertainty, since idiosyncratic uncertainty is now embedded in ωt,j ,

ωt,j =(βθ)j

Yt+j

Ct+jΩε−1

t,j

Ft, Ft = Et

∞∑l=0

(βθ)lYt+l

Ct+lΩε−1

t,l

Ωt,j =

πt+j πt+j−1 · · · πt+1, j ≥ 1

1, j = 0., πt ≡

Pt

Pt−1.

Note that if θ = 0, then ωt,0 = 1, ωt,j = 0, for j > 0, in which case,

Pt =ε

ε− 1MCt .

Scaling the Marginal Price Setter’s Price

Let

st ≡MCt

Pt=

(1− ν) Wt

Pt

eat= (1− ν) eτtCtN

ϕt /e

at .

Denoting pt ≡ Pt/Pt :

pt =Kt

Ft

where

Kt = Et

∞∑j=0

(βθ)jYt+j

Ct+jΩε

t,j

ε

ε− 1st+j

=ε

ε− 1

Yt

Ct

(1− ν) eτtCtNϕt

eat+ βθEt π

εt+1Kt+1 (1)

Ft = Et

∞∑l=0

(βθ)lYt+l

Ct+lΩε−1

t,l =Yt

Ct+ βθEt π

ε−1t+1 Ft+1 (2)

derive

Moving On to Aggregate Restrictions

Link between aggregate price level, Pt , and Pi,t , i ∈ [0, 1].

I Potentially complicated because there are MANY prices, Pi,t , i ∈ [0, 1].

I Important: Calvo result.

Link between aggregate output, Yt , and aggregate employment, Nt .

I Complicated, because Yt depends not just on Nt but also on how employment

is allocated across sectors.

I Important: Tack Yun distortion.

Market clearing conditions.

I Bond market clearing.

I Labor and goods market clearing.

Inflation and Marginal Price Setter

Calvo result: derive

Pt =(

(1− θ) P1−εt + θP1−ε

t−1

) 11−ε

Divide by Pt :

1 =

((1− θ) p1−ε

t + θ

(1

πt

)1−ε) 1

1−ε

pt is relative price of marginal price setter. Then,

pt =

[1− θ (πt)

ε−1

1− θ

] 11−ε

Tack Yun Distortion (JME1996)

Define Y ∗t :

Y ∗t ≡∫ 1

0

Yi,tdi

(=

∫ 1

0

eatNi,tdi = eatNt

)demand curve︷︸︸︷

= Yt

∫ 1

0

(Pi,t

Pt

)−εdi = YtP

εt

∫ 1

0

(Pi,t)−ε di

= YtPεt (P∗t )−ε .

So,

Yt = p∗t Y∗t , p

∗t =

(P∗tPt

)ε= ‘Tack Yun Distortion’

Then (brilliant!):

Yt = p∗t eatNt .

Understanding the Tack Yun Distortion

Relationship between aggregate inputs and outputs:

Yt = p∗t eatNt .

Note that p∗t is a function of the ratio of two averages (with different

weights) of Pi,t , i ∈ (0, 1) :

p∗t =

(P∗tPt

)ε,

where

P∗t =

(∫ 1

0

P−εi,t di

)−1ε

, Pt =

(∫ 1

0

P(1−ε)i,t di

) 11−ε

The Tack Yun distortion, p∗t , is a measure of dispersion in prices, Pi,t ,

i ∈ [0, 1] .

Understanding the Tack Yun Distortion

Why is a ratio of two different weighted averages of prices a measure of

dispersion?

I Example

x

x=

12x1 + 1

2x2

14x1 + 3

4x2

=

1 if x1 = x2

6= 1 x1 6= x2.

But, the Tack Yun distortion is not the ratio of just any two different

weighted averages.

I In fact, simple Jensen’s inequality argument shows: proof

p∗t ≤ 1, with equality iff Pi,t = Pj,t for all i , j .

I Actually, it must be that proof

Yt =

average of concave functions of Yi,t , i ∈ [0, 1]︷ ︸︸ ︷∫ 1

0

Yε−1ε

i,t dj

ε

ε−1

≤ eatNt .

Law of Motion of Tack Yun Distortion

We have, using the Calvo result:

P∗t =[(1− θ) P−εt + θ

(P∗t−1

)−ε]−1ε

Dividing by Pt :

p∗t ≡(P∗tPt

)ε=

[(1− θ) p−εt + θ

πεtp∗t−1

]−1

=

(1− θ)

[1− θ (πt)

ε−1

1− θ

] −ε1−ε

+ θπεtp∗t−1

−1

(4)

using the restriction between pt and aggregate inflation developed earlier.

Market Clearing

We now summarize the market clearing conditions of the model.

Labor, bond and goods markets.

Other Market Clearing Conditions

Bond market clearing:

Bt+1 = 0, t = 0, 1, 2, ...

Labor market clearing:

supply of labor︷︸︸︷Nt =

demand for labor︷ ︸︸ ︷1∫

0

Ni,tdi

Goods market clearing:

demand for final goods︷ ︸︸ ︷Ct + Gt =

supply of final goods︷︸︸︷Yt ,

and, using relation between Yt and Nt :

Ct + Gt = p∗t eatNt (6)

Equilibrium Conditions

6 equations in 7 unknowns: Ct , p∗t ,Ft ,Kt ,Nt ,Rt , πt , and two policy variables:

Kt =ε

ε− 1

Yt

Ct

(1− ν) eτtCtNϕt

At+ βθEt π

εt+1Kt+1 (1)

Ft =Yt

Ct+ βθEt π

ε−1t+1 Ft+1 (2),

Kt

Ft=

[1− θπ(ε−1)

t

1− θ

] 11−ε

(3)

p∗t =

(1− θ)

(1− θπ(ε−1)

t

1− θ

) εε−1

+θπεtp∗t−1

−1

(4)

1

Ct= βEt

1

Ct+1

Rt

πt+1(5), Ct + Gt = p∗t e

atNt (6)

System underdetermined! Flexible price case, θ = 0 is interesting.

Classical Dichotomy Under Flexible Prices

Classical Dichotomy : when prices flexible, θ = 0, then real variables

determined.

I Equations (2),(3) imply:

Ft = Kt =Yt

Ct,

which, combined with (1) implies

ε (1− ν)

ε− 1×

Marginal Cost of work︷ ︸︸ ︷eτtCtN

ϕt =

marginal benefit of work︷︸︸︷eat

I Expression (6) with p∗t = 1 (since θ = 0) is

Ct + Gt = eatNt .

Thus, we have two equations in two unknowns, Nt and Ct .

Classical Dichotomy: No Uncertainty

Real interest rate, R∗t ≡ Rt/πt+1, is determined:

R∗t =1Ct

β 1Ct+1

.

So, with θ = 0, the following are determined:

R∗t ,Ct ,Nt , t = 0, 1, 2, ...

What about the nominal variables?

I Suppose the central bank wants a given sequence of inflation rates, πt ,

t = 0, 1, ... .

I Then it must produce the following sequence of interest rates:

Rt = πt+1R∗t , t = 0, 1, 2, ...

How Does the CB Set the Interest Rate?

When NK model leaves out money demand, implicitly author has in mind

that money enters preferences additively separably:

maxE0

∞∑t=0

βt

(logCt − exp (τt)

N1+ϕt

1 + ϕ+ γlog

(Mt+1

Pt

)),

s.t. PtCt + Bt+1 + Mt+1

≤WtNt + Rt−1Bt + Mt

+Profits net of government transfers and taxest ,

where Mt+1 is the beginning of period t + 1 stock of money.

Labor and bond first order conditions same as before.

Money first order condition: proof

Mt+1

Pt=

(Rt

Rt − 1

)γCt ,

which looks like a standard undergrad money demand equation.

Classical Dichotomy versus New Keynesian Model

When θ = 0, then the Classical Dichotomy occurs.

I In this case, Central Bank cannot affect R∗t ,Ct ,Nt .

I Monetary policy simply affects the split in the real interest rate between

nominal and real rates:

R∗t =Rt

πt+1.

I For a careful treatment when there is uncertainty, see.

When θ > 0 (NK model) then cannot pin down any of the 7 endogenous

variables using the 6 available equations.

I In this case, monetary policy matters for R∗t ,Ct ,Nt .

Monetary Policy in New Keynesian Model

Suppose θ > 0, so that we’re in the NK model and monetary policy matters.

The standard assumption is that the monetary authority sets money growth to

achieve an interest rate target, and that that target is a function of inflation:

Rt/R = (Rt−1/R)α exp (1− α) [φπ(πt − π) + φxxt ] (7)’,

where xt denotes the log deviation of actual output from target (more on this

later).

This is a Taylor rule, and it satisfies the Taylor Principle when φπ > 1.

Smoothing parameter: α.

I Bigger is α the more persistent are policy-induced changes in the interest rate.

Equilibrium Conditions of NK Model with Taylor Rule

Kt =ε

ε− 1

Yt

Ct

(1− ν) eτtCtNϕt

At+ βθEt π

εt+1Kt+1 (1)

Ft =Yt

Ct+ βθEt π

ε−1t+1 Ft+1 (2),

Kt

Ft=

[1− θπ(ε−1)

t

1− θ

] 11−ε

(3)

p∗t =

(1− θ)

(1− θπ(ε−1)

t

1− θ

) εε−1

+θπεtp∗t−1

−1

(4)

1

Ct= βEt

1

Ct+1

Rt

πt+1(5), Ct + Gt = p∗t e

atNt (6)

Rt/R = (Rt−1/R)α exp (1− α) [φπ(πt − π) + φxxt ] (7).

Natural Equilibrium

When θ = 0, then

ε (1− ν)

ε− 1×

Marginal Cost of work︷ ︸︸ ︷eτtCtN

ϕt =

marginal benefit of work︷︸︸︷eat

so that we have a form of efficiency when ν is chosen to that

ε (1− ν) / (ε− 1) = 1.

In addition, recall that we have allocative efficiency in the flexible price

equilibrium.

So, the flexible price equilibrium with the efficient setting of ν represents a

natural benchmark for the New Keynesian model, the version of the model in

which θ > 0.

I We call this the Natural Equilibrium.

To simplify the analysis, from here on we set Gt = 0.

Natural Equilibrium

With Gt = 0, equilibrium conditions for Ct and Nt :

Marginal Cost of work︷ ︸︸ ︷eτtCtN

ϕt =

marginal benefit of work︷︸︸︷eat

aggregate production relation: Ct = eatNt .

Substituting,

eτt eatN1+ϕt = eat → Nt = exp

(−τt

1 + ϕ

)Ct = exp

(at −

τt1 + ϕ

)R∗t =

1Ct

βEt1

Ct+1

=1

βEtCt

Ct+1

=1

βEtexp(−∆at+1 + ∆τt+1

1+ϕ

)

Natural Equilibrium, cnt’d

Natural rate of interest:

R∗t =1Ct

βEt1

Ct+1

=1

βEtexp(−∆at+1 + ∆τt+1

1+ϕ

)Two models for at :

DS : ∆at+1 = ρ∆at + εat+1

TS : at+1 = ρat + εat+1

Model for τt :

τt+1 = λτt + ετt+1

Natural Equilibrium, cnt’d

Suppose the εt ’s are Normal. Then,

Etexp

(−∆at+1 +

∆τt+1

1 + ϕ

)= exp

(−Et∆at+1 + Et

∆τt+1

1 + ϕ+

1

2V

)

V = σ2a +

σ2τ

(1 + ϕ)2

Then, with r∗t ≡ logR∗t : r∗t = − log β + Et∆at+1 − Et∆τt+1

1+ϕ −12V .

Useful: consider how natural rate responds to εat shocks under DS and TS

models for at and how it responds to ετt shocks.

I To understand how r∗t responds, consider implications of consumption

smoothing in absence of change in r∗t .

I Hint: in natural equilibrium, r∗t steers the economy so that natural equilibrium

paths for Ct and Nt are realized.

Conclusion

Described NK model and derived equilibrium conditions.

I The usual version of model represents monetary policy by a Taylor rule.

When θ = 0, so that prices are flexible, then monetary policy has no impact

on Ct ,Nt ,R∗t .

I Changes in money growth move prices and wages in such a way that real

wages do not change and employment and output don’t change.

When prices are sticky, then a policy-induced reduction in the interest rate

encourages more nominal spending.

I The increased spending raises Wt more than Pt because of the sticky prices,

thereby inducing the increased labor supply that firms need to meet the extra

demand.I Firms are willing to produce more goods because:

F The model assumes they must meet all demand at posted prices.

F Firms make positive profits, so as long as the expansion is not too big they still

make positive profits, even if not optimal.

Household Intertemporal Euler Equation

Equation (5), the household’s intertemporal Euler equation, may seem

counterintuitive because (5) has Et in it, while the household’s objective has

E0 in it.

To explain the apparent inconsistency, it is necessary to be more explicit

about expectations and uncertainty.

To this end, let

st − ‘realization of uncertainty in period t’ (in model, st =[

at τt

]′).

st − ‘history of exogenous shocks up to period t′, st ≡ (s0, s1, ..., st) =(st−1, st

)I You may find it useful to draw an ‘event tree’ conditional on a particular s0,

describing all possible histories, s t , for t = 1, 2, 3 in the case where st is a

scalar and st ∈sh, s l

.

Some Simple Properties of Probabilities

Let µ (st) denote the probability of history, st , and assume µ (st) > 0.

I Thus,∑

stµ(s t)

= 1, where∑

st means ‘sum, over all possible histories, s t ,

for given t’.

Also, st+1|st means ‘all possible histories, st+1, that are consistent with a

given history, st .

Following is a property of any two random vectors, x , y :

µ (x |y) = µ (x , y) /µ (y) .

I In words, ‘the conditional probability of x given y is the joint probability of x

and y , divided by the marginal density of y .

I The marginal density of y , µ (y) ≡∑

x µ (x , y) , where∑

x denotes ‘sum over

all possible values of x ’.

The Lagrangian Representation of the Household Problem,

in State Notation

It is convenient to index variables by st rather than by t.

The household problem, in Lagrangian form:

maxc(st),B(st),N(st)

∞∑t=0

βt∑st

µ(st)u(c(st),N(st))

+υ(st)

[W(st)N(st)

+ B(st−1

)R(st−1

)+gov’t taxes and profits - P

(st)c(st)− B

(st)]

I υ(s t)

˜ Lagrange multiplier on household budget constraint in state s t . (‘The

value of one unit of currency in state s t ’.)

I For intuition, verify that the following must be true: υ(s t)> 0. (Hint: if you

set υ(s t)

= 0, the solution is c(s t)

=∞ and N(s t)

tiny, violating the budget

constraint and making the expression in square brackets −∞.)

I Here, R(s t), B(s t)

correspond to Rt , Bt+1, respectively.

Household Problem

Household:

maxc(st),B(st),N(st)

∞∑t=0

βt∑st

µ(st)u(c(st),N(st))

+υ(st)

[W(st)N(st)

+ B(st−1

)R(st−1

)+gov’t taxes and profits - P

(st)c(st)− B

(st)]

First order conditions associated with c (st) and B (st) needed to explain 5.

I Fonc, c(s t), for any specific value of s t :

uc(c(s t),N(s t))

= υ(s t)P(s t)

I Fonc, B(s t), for given s t :

βtµ(s t)υ(s t)

= βt+1∑

st+1|stµ(s t+1

)υ(s t+1

)R(s t)

Household Problem

Fonc, c (st), for any specific value of st :

uc(c(st),N(st))

= υ(st)P(st)

Fonc, B (st) , for given st :

βtµ(st)υ(st)

= βt+1∑

st+1|stµ(st+1

)υ(st+1

)R(st)

Note that B (st) appears in two places: (i) in state st when you buy it (this

cost in st is on left of equality); and (ii) in states (st , st+1) for each st+1

when the bond pays off. In each st+1|st the bond pays off the same amount,

R (st) .

Substitute and rearrange:

uc(c(s t),N(s t))

= β∑

st+1|st

µ(s t+1

)µ (s t)

uc(c(s t+1

),N(s t+1

)) R(s t)

P (s t+1) /P (s t).

Household Problem

So, we have the following intertemporal Euler equation:

uc(c(s t),N(s t))

= β∑

st+1|st

µ(s t+1

)µ (s t)

uc(c(s t+1

),N(s t+1

)) R(s t)

P (s t+1) /P (s t),

or, using the property of conditional distributions:

uc(c(s t),N(s t))

= β∑

st+1|stµ(s t+1|s t

)uc(c(s t+1

),N(s t+1

)) R(s t)

P (s t+1) /P (s t),

which is exactly equation 5 in more precise notation.

Go Back

Time t Marginal Utility of a Unit of Currency in the Future

We simplify the analysis by temporarily ignoring uncertainty.

By writing the household problem in Lagrangian form, it is easy to verify that

βjυt+j is the time t marginal utility of one unit of currency in period t + j :

βjυt+j = βj uc,t+j

Pt+j,

where uc,t denotes the derivative of period t utility with respect to

consumption.

I To understand the above expression, with one unit of currency in period t + j

one can obtain 1/Pt+j units of consumption goods in that period, which has

utility value, uc,t+j/Pt+j , in t + j .

I Then, multiply by βj to convert into period t utility units.

The Intermediate Good Firm’s Objective is Equivalent to

the Discounted Present Value of Profits

It is interesting to note that the intertemporal Euler equation implies βjυt+j

in effect discounts period t + j cash payments to the owners of firms by the

nominal interest rate.

To see this, repeatedly substitute out the future marginal utility of

consumption using the Euler equation:

I uc,t = βuc,t+1RtPt/Pt+1 = β2uc,t+2RtPt/Pt+1Rt+1Pt+1/Pt+2 =

β2uc,t+2RtRt+1Pt/Pt+2

I Again,

uc,t = β3uc,t+3RtRt+1Rt+2Pt/Pt+3.

I Continuing,

uc,t = βjuc,t+jRtRt+1 · · · Rt+j−1Pt/Pt+j .

So,

βj uc,t+j

Pt+j=

uc,tPt

1

RtRt+1 · · · Rt+j−1

The Firm’s Objective

We conclude

E it

∞∑j=0

βj υt+j [Pi,t+jYi,t+j − Pt+jst+jYi,t+j ]

=uc,tPt

E it

∞∑j=0

Pi,t+jYi,t+j − Pt+jst+jYi,t+j

RtRt+1 · · · Rt+j−1

The objective of the firm is the present discounted value (using the nominal

rate of interest to do the discounting) of money profits.

Once the flow of future money payments is converted into a time t money

value, that value is converted into real terms by multiplying by uc,t/Pt .

I This last step makes no difference, since whether one maximizes a money

measure of profits or a utility measure of profits is the same.

Go Back

Review of Marginal Cost

Here is a more careful derivation of the marginal cost formula in the handout.

I Assume there are n inputs to production

I Production function is linear homogeneous.

Suppose the production function is

Y = F (x1, x2, ..., xn) ,

where F is linear homogeneous:

λY = F (λx1, λx2, ..., λxn) , all λ > 0.

Total derivative property of linear homogeneity:

Y =n∑

i=1

Fi (λx1, λx2, ..., λxn) xi , all λ > 0.

Review of Marginal Cost

The firm is assumed to be competitive in input markets, where pj is given,

j = 1, ..., n.

The cost of producing a given quantity of output, Y , denoted C (Y ) , is

C (Y ) = minxi ,all i

p1x1 + ...+ pnxn,

subject to the production function.

Letting λ ≥ 0 denote the multiplier on the constraint, we have

C (Y ) = minxi ,all i

p1x1 + ...+ pnxn + λ [Y − F (x1, x2, ..., xn)] . (1)

Marginal Cost

The n + 1 conditions for an optimum are (1), plus

λ =pjFj, j = 1, ..., n,

where Fj denotes the derivative of F with respect to its j th argument.

Denote the n + 1 variables, λ, xj , j = 1, ..., n, which solve the n + 1 conditions

by λ∗ (Y ) , x∗j (Y ) , j = 1, ..., n. (It is convenient to make the dependence on

Y explicit, although the solutions also depend on the pj ’s.)

Then,

C (Y ) = p1x∗1 (Y ) + ...+ pnx

∗n (Y ) + λ∗ (Y ) [Y − F (x∗1 (Y ) , ..., x∗n (Y ))]

= λ∗ (Y )Y ,

by total derivative property of linear homogeneity.

Marginal Cost

The cost function:

C (Y ) = p1x∗1 (Y ) + ...+ pnx

∗n (Y ) + λ∗ (Y ) [Y − F (x∗1 (Y ) , ..., x∗n (Y ))] .

Marginal cost is the derivative of C with respect to Y :

C ′ (Y ) = λ∗ (Y ) + [Y − F (x∗1 (Y ) , ..., x∗n (Y ))]dλ∗ (Y )

dY

+n∑

j=1

[pj − λ∗ (Y )Fj (x∗1 (Y ) , ..., x∗n (Y ))]dx∗j (Y )

Y

= λ∗ (Y ) ,

because the optimality conditions imply that the n + 1 objects in square

brackets are zero.

We conclude that marginal cost corresponds to pj/Fj for j = 1, ..., n. Go Back

Calvo versus Rotemberg

“Why don’t we just do Rotemberg adjustment costs? It’s much easier and

gives the same reduced form anyway”.

One answer:I When people do Rotemberg adjustment costs, they are implicitly actually

doing Calvo.

F Rotemberg has an coefficient, φ, on the price adjustment cost term,

(Pt − Pt−1)2, and it is hard to think about what is an empirically plausible

value for it. So, in practice, when you do Rotemberg (whether estimating or

calibrating) you have to convert to Calvo to evaluate the value of φ that you

are using.

F Why? From the perspective of Calvo, the parameter φ is a function of the

Calvo parameter, θ, and that is something that people have strong views about

because it can be directly estimated from observed micro data.

I Similarity of Calvo and Rotemberg only reflects that people have the habit of

linearizing around zero inflation. Different in empirically plausible case of

inflation. Difference is small in the simple model, but not in more plausible

models.

Calvo versus Rotemberg

Rotemberg is completely against the Zeitgeist of modern macroeconomics.

Modern macro is increasingly going to micro data (see, for example, HANK)

to look for guidance about how to build macro models.

I Another example (besides HANK) is the finding that the network nature of

production (see Christiano et. al. 2011 and Christiano (2016)) matters for key

properties of the New Keynesian model, including (i) the cost of inflation, (ii)

the slope of the Phillips curve and (iii) the value of the Taylor Principle for

stabilizing inflation. This is a growing area of research in macroeconomics.

I Another example is the importance of networks in financial firms for the

possibility of financial crisis.

Calvo’s interesting implications for the distribution of prices in micro data has

launched an enormous literature (see Eichenbaum, et al, Nakamura and

Steinsson and many more papers). It is generating a picture of what kind of

model is needed to eventually replace the Calvo model. Go Back

Firms that Can Change Price at t

Let Pt denote the price set by the 1− θ firms who optimize at time t.

Expected value of future profits sum of two parts:

I future states in which price is still Pt , so Pt matters.

I future states in which the price is not Pt , so Pt is irrelevant.

That is,

E it

∞∑j=0

βjυt+j [Pi,t+jYi,t+j − Pt+jst+jYi,t+j ]

=

Zt︷ ︸︸ ︷Et

∞∑j=0

(βθ)j υt+j

[PtYi,t+j − Pt+jst+jYi,t+j

]+Xt ,

where

I Zt is the present value of future profits over all future states in which the

firm’s price is Pt .

I Xt is the present value over all other states, so dXt/dPt = 0.

Decision By Firm that Can Change Its Price

Substitute out demand curve:

Et

∞∑j=0

(βθ)j υt+j

[PtYi,t+j − Pt+jst+jYi,t+j

]= Et

∞∑j=0

(βθ)j υt+jYt+jPεt+j

[P1−εt − Pt+jst+j P

−εt

].

Differentiate with respect to Pt :

Et

∞∑j=0

(βθ)j υt+jYt+jPεt+j

[(1− ε)

(Pt

)−ε+ εPt+jst+j P

−ε−1t

]= 0,

→ Et

∞∑j=0

(βθ)j υt+jYt+jPε+1t+j

[Pt

Pt+j− ε

ε− 1st+j

]= 0.

I When θ = 0, get standard result - price is fixed markup over marginal cost.

Decision By Firm that Can Change Its Price

Substitute out the multiplier:

Et

∞∑j=0

(βθ)j

= υt+j︷ ︸︸ ︷u′ (Ct+j)

Pt+jYt+jP

ε+1t+j

[Pt

Pt+j− ε

ε− 1st+j

]= 0.

I Using assumed log-form of utility,

Et

∞∑j=0

(βθ)jYt+j

Ct+j(Xt,j)

−ε[ptXt,j −

ε

ε− 1st+j

]= 0,

pt ≡Pt

Pt, πt ≡

Pt

Pt−1, Xt,j =

1

πt+j πt+j−1···πt+1, j ≥ 1

1, j = 0.,

’recursive property’: Xt,j = Xt+1,j−11

πt+1, j > 0

Decision By Firm that Can Change Its Price

Want pt in:

Et

∞∑j=0

(βθ)jYt+j

Ct+j(Xt,j)

−ε[ptXt,j −

ε

ε− 1st+j

]= 0

Solving for pt , we conclude that prices are set as follows:

pt =Et

∑∞j=0 (βθ)j

Yt+j

Ct+j(Xt,j)

−ε εε−1 st+j

Et

∑∞j=0

Yt+j

Ct+j(βθ)j (Xt,j)

1−ε =Kt

Ft.

Need convenient expressions for Kt , Ft .

Decision By Firm that Can Change Its Price

Recall,

pt =Et

∑∞j=0 (βθ)j

Yt+j

Ct+j(Xt,j)

−ε εε−1 st+j

Et

∑∞j=0 (βθ)j

Yt+j

Ct+j(Xt,j)

1−ε =Kt

Ft

The numerator has the following simple representation:

Kt = Et

∞∑j=0

(βθ)jYt+j

Ct+j(Xt,j)

−ε ε

ε− 1st+j

=ε

ε− 1

Yt

Ct

(1− ν) eτtCtNϕt

eat+ βθEt

(1

πt+1

)−εKt+1 (1),

after using st = (1− ν) eτtCtNϕt /e

at .

Similarly,

Ft =Yt

Ct+ βθEt

(1

πt+1

)1−ε

Ft+1 (2)

Simplifying Numerator

Kt = Et

∞∑j=0

(βθ)jYt+j

Ct+j(Xt,j)

−ε ε

ε− 1st+j

=ε

ε− 1

Yt

Ctst

+ βθEt

∞∑j=1

Yt+j

Ct+j(βθ)j−1

=Xt,j , recursive property︷ ︸︸ ︷

1

πt+1Xt+1,j−1

−ε

ε

ε− 1st+j

=ε

ε− 1

Yt

Ctst + Zt ,

where

Zt = βθEt

∞∑j=1

(βθ)j−1 Yt+j

Ct+j

(1

πt+1Xt+1,j−1

)−εε

ε− 1st+j

Simplifying Numerator, cnt’d

Kt = Et

∞∑j=0

(βθ)jYt+j

Ct+j

(Xt,j

)−ε ε

ε− 1st+j =

ε

ε− 1st + Zt

Zt = βθEt

∞∑j=1

(βθ)j−1 Yt+j

Ct+j

(1

πt+1Xt+1,j−1

)−ε ε

ε− 1st+j

= βθEt

(1

πt+1

)−ε ∞∑j=0

(βθ)jYt+j+1

Ct+j+1X−εt+1,j

ε

ε− 1st+1+j

= βθ

=Et by LIME︷ ︸︸ ︷EtEt+1

(1

πt+1

)−ε ∞∑j=0

(βθ)jYt+j+1

Ct+j+1X−εt+1,j

ε

ε− 1st+1+j

= βθEt

(1

πt+1

)−ε exactly Kt+1!︷ ︸︸ ︷Et+1

∞∑j=0

(βθ)jYt+j+1

Ct+j+1X−εt+1,j

ε

ε− 1st+1+j

Unscaled Marginal Price Setter’s Price

Optimal price, Pt , of marginal price setter

Pt = Pt

Et

∑∞j=0 (βθ)j

Yt+j

Ct+j(Xt,j)

−ε εε−1

=MCt+j/Pt+j︷︸︸︷st+j

Et

∑∞j=0 (βθ)j

Yt+j

Ct+j(Xt,j)

1−ε

=Et

∑∞j=0 (βθ)j

Yt+j

Ct+j(Xt,j)

1−ε εε−1MCt+j

Et

∑∞j=0 (βθ)j

Yt+j

Ct+j(Xt,j)

1−ε

= Et

∞∑j=0

≡ωt,j︷ ︸︸ ︷ (βθ)jYt+j

Ct+j(Xt,j)

1−ε

Et

∑∞l=0 (βθ)l Yt+l

Ct+l(Xt,l)

1−ε

ε

ε− 1MCt+j

Go Back

Aggregate Price Index: Calvo Result

Trick: rewrite the aggregate price index.

I let p ∈ (0,∞) the set of logically possible prices for intermediate good

producers.

I let gt (p) ≥ 0 denote the measure (e.g., ’number’) of producers that have

price, p, in t

I let gt−1,t (p) ≥ 0, denote the measure of producers that had price, p, in t − 1

and could not re-optimize in t

I Then,

Pt =

(∫ 1

0

P(1−ε)i,t di

) 11−ε

=

(∫ ∞0

gt (p) p(1−ε)dp

) 11−ε

.

Note:

Pt =

((1− θ) P1−ε

t +

∫ ∞0

gt−1,t (p) p(1−ε)dp

) 11−ε

.

Aggregate Price Index: Calvo Result

Calvo randomization assumption:

measure of firms that had price, p, in t−1 and could not change︷ ︸︸ ︷gt−1,t (p)

= θ ×

measure of firms that had price p in t−1︷ ︸︸ ︷gt−1 (p)

Aggregate Price Index: Calvo Result

Using gt−1,t (p) = θgt−1 (p) :

Pt =

((1− θ) P1−ε

t +

∫ ∞0

gt−1,t (p) p(1−ε)dp

) 11−ε

Pt =

(1− θ) P1−εt + θ

=P1−εt−1︷ ︸︸ ︷∫ ∞

0

gt−1 (p) p(1−ε)dp

1

1−ε

This is the Calvo result:

Pt =(

(1− θ) P1−εt + θP1−ε

t−1

) 11−ε

Wow, simple!: Only two variables: Pt and Pt−1. Go Back

Tack Yun Distortion

Let f (x) = x4, a convex function. Then,

convexity: αx41 + (1− α) x4

2 > (αx1 + (1− α) x2)4

for x1 6= x2, 0 < α < 1.

Applying this idea:

convexity:

∫ 1

0

(P

(1−ε)i,t

) εε−1

di ≥(∫ 1

0

P(1−ε)i,t di

) εε−1

⇐⇒(∫ 1

0

P−εi,t di

)≥(∫ 1

0

P(1−ε)i,t di

) εε−1

⇐⇒

P∗t︷ ︸︸ ︷(∫ 1

0

P−εi,t di

)−1ε

≤

Pt︷ ︸︸ ︷(∫ 1

0

P(1−ε)i,t di

) 11−ε

Go Back

Efficient Sectoral Allocation of Resources

Consider the following problem

maxNi,t ,i∈[0,1]

[∫ 1

0

(eatNi,t)ε−1ε di

] εε−1

subject to a given amount of total employment:

Nt =

∫ 1

0

Ni,tdi

In Lagrangian form:

maxNi,t ,i∈[0,1]

Yt︷ ︸︸ ︷[∫ 1

0

(eatNi,t)ε−1ε di

] εε−1

+λ

[Nt −

∫ 1

0

Ni,t

],

where λ ≥ 0 denotes the Lagrange multiplier.

Efficient Sectoral Allocation of Resources

Lagrangian problem:

maxNi,t ,i∈[0,1]

Yt︷ ︸︸ ︷[∫ 1

0

(eatNi,t

) ε−1ε di

] εε−1

+λ

[Nt −

∫ 1

0Ni,t

],

where λ ≥ 0 denotes the Lagrange multiplier.

First order necessary condition for optimization:(Yt

Ni,t

) 1ε

(eat )ε−1ε = λ→ Ni,t = Nj,t = Nt , for all i , j ,

so Yt is as big as it possibly can be for given aggregate employment, when

Yt =

[∫ 1

0

(eatNi,t

) ε−1ε di

] εε−1

= eatNt .

Result is obvious because(Ni,t

) ε−1ε is strictly concave in Ni,t .

SupposeLaborNot AllocatedEqually

• Example:

• Notethatthisisaparticulardistributionoflaboracrossactivities:

Nit 2)Nt i 0, 12

21 " ) Nt i 12 , 1

, 0 t ) t 1.

;0

1Nitdi 1

2 2)Nt 12 21 " ) Nt Nt

LaborNot AllocatedEqually,cnt’dYt ;

0

1Yi,t

/"1/ di

//"1

;0

12 Yi,t

/"1/ di ;

12

1Yi,t

/"1/ di

//"1

eat ;0

12 Ni,t

/"1/ di ;

12

1Ni,t

/"1/ di

//"1

eat ;0

12 2)Nt

/"1/ di ;

12

121 " ) Nt

/"1/ di

//"1

eatNt ;0

12 2)

/"1/ di ;

12

121 " )

/"1/ di

//"1

eatNt 12 2) /"1/ 12 21 " )

/"1/

//"1

eatNtf)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.88

0.9

0.92

0.94

0.96

0.98

1

f

Efficient Resource Allocation Means Equal Labor Across All Sectors

6

10

f 12 2

1 1

2 21 1

1

Go Back

Money Demand

The Lagrangian representation of the household problem is (λt ≥ 0 is multiplier):

maxE0

∞∑t=0

βt(

log Ct − exp (τt)N1+ϕt

1 + ϕ+ γlog

(Mt+1

Pt

))+λt (WtNt + Rt−1Bt + Mt − PtCt − Bt+1 −Mt+1).

First order conditions:

Ct : u′ (Ct) = λtPt ; Bt+1 : λt = βEtλt+1Rt

Mt+1 :λt =γ

Mt+1+ βEtλt+1

I Substitute out for βEtλt+1 in Mt+1 equation from Bt+1 equation; then substitute out

for λt from Ct equation and rearrange, to get

Mt+1

Pt=

(Rt

Rt − 1

)γCt .

Go Back